Observational rather than from designed experiments.
Calibration to connect observations to physics.
Sparsity is inevitable.
4a. Objects are mixed: different life stages and time-scale of evolution.
4b. Time-scales are much larger than we can observe.
Astrophysics: a parsimonious mathematical representation of expected signal from a physical process that generates an emission detectable by instruments.
Statistics: a stochastic representation of the data-generating process that also accounts for the discrepancy between the astrophysical model and the data.
Use \(\mathrm{Poisson}(g(\theta))\) as a statistical model.
Astrophysical model has \(g(\theta)\) as a power-law.
Uncertainty quantification is at the heart of statistical model.
Model misspecification can cause spurious results.
Different model fits on the same data can reveal completely different possibilities.
Don’t blindly make inference on the highest mode of the posterior distribution!
The better the statistical and astronomical data reflect the data the better quality of what the data reveal to us.
Carefully consider implicit modelling and statistical assumptions; these will affect scientific findings.
all models are wrong but some are useful Box & Draper (1987)
All data have stories, but some are mistold.
All assumptions are meant to be helpful, but some can be harmful.
All prior distributions are informative, even those that are uniform.
All models are subject to interpretation, but some are less contrived.
All statistical tests have thresholds, but some are mis-set.
All model checks consider variations of the data, but some variants are more relevant than others.
::::
Data is often collected for a specific purpose.
These are not randomly and uniformly selected data!
Characteristics of individual surveys can affect overall interpretation when combined together.
Preprocessing can introduce statistical and systematic errors.
Calibration measurements are imperfect since these themselves include measurement errors and systematics.
\(\chi^2\) minimisation assumes Gaussian approximation of measurement errors.
Using a Gaussian approximation of Poisson count data is not valid when the variance is very different from the average count.
try fitting counts directly and use a different fit statistic.
\[\log(X) \sim \mathrm{Unif}[a,b]\]
is not non-informative on the original scale of \(X\).
bounded uniform priors must be used with care because these completely exclude portions of the parameter space.
set uniform bounds wide enough to not influence the likelihood.
“best to start with the physics and then consider whether the empirical findings make sense in terms of the physics and/or how we can make sense of them.”